Analytical Filters

Low-Pass Spectral Graph Filter

The low-pass filter is a refinable, self-similar rational function useful for signal smoothing. The scale parameter, \(s\), controls the cutoff frequency.

\[\phi_s(\mathbf{\Lambda}) = \frac{I}{s\mathbf{\Lambda} + I}\]

High-Pass Spectral Graph Filter

The high-pass filter isolates high-frequency variations on the graph.

\[\mu_s(\mathbf{\Lambda}) = \frac{s\mathbf{\Lambda}}{s\mathbf{\Lambda} + I}\]

Band-Pass Spectral Graph Filter

A convenient closed-form wavelet generating kernel that serves as an alternative to custom-designed filters. The scale \(s\) adjusts the center frequency of the band.

\[\Psi_s(\mathbf{\Lambda}) = \frac{4s\mathbf{\Lambda}}{(s\mathbf{\Lambda}+I)^2}\]

This filter qualifies as a wavelet generating kernel for the SGWT, since \(\Psi(0)=0\) and the admissibility condition is satisfied. The admissibility constant of this band-pass filter is \(C_{\Psi}=8/3\).

\[\Psi(0)=0 \qquad\text{and}\quad \int_0^{\infty}\dfrac{\Psi^2(x)}{x}\mathrm{d}x <\infty\]